Symmetry-enforced electronic nodal straight lines in CsNb3SBr7

We propose the quaternary-compounds CsNb3SBr7 is a nodal-straight-line semimetal candidate based on the first-principles calculations and symmetry analyses. There are a pair of nodal straight lines locate in the k z = 0 plane of Brillouin zone, which is protected by the crystal symmetry. The topological properties of nodal-straight-line state are characterized by the nontrivial Berry phase and Berry curvature. On the (001) surface of CsNb3SBr7, Fermi arcs form the belt-like surface state, connecting the nodal straight lines with opposite chirality. Furthermore, the Hofstadter’s butterfly and optical conductivity are investigated using a slab sample. These results not only reveal the symmetric protection mechanism of nodal straight lines, but also pave a way for exploring the electronic and optical properties of CsNb3SBr7 in further condensed matter physics experiments.


Introduction
Topological nodal line quantum states [1][2][3][4][5][6][7][8][9][10][11][12][13] of solid materials with unique electronic and phononic properties have triggered tremendous interest in condensed matter physics and material science [14][15][16][17][18][19]. In recent years, great efforts have been made in search of topological semimetal materials which are featured with nontrivial band degeneracy in momentum space. The novel topological phases exhibit a rich classification in geometry of gapless points, such as nodal rings [20,21], nodal chains [22][23][24], and nodal net [25,26]. They immensely enrich our knowledge of topological physics. In electronic systems, nodal line states usually are characterized by the band-touching points of conduction and valence bands near Fermi energy. Its important signature is the drum-head like surface state, which is flat in energy and filled inside the projection of the closed nodal line [7]. The intriguing drum-head like surface state provides an important platform to realize the strong electronic correlation effect and superconductivity [17][18][19]. Nowadays, thousands of nodal line semimetals have been predicted, and some of them are discovered in the experiment [27][28][29][30][31]. These works compose the topological material database [32][33][34][35] and greatly promote the studies of physical properties of topological materials.
Very recently, many phononic nodal-straight-line states have been reported in solid materials [36][37][38]. As is known, the existence of nodal line states is intimately associated with crystal symmetries. After investigating the conditions under which nodal straight lines occur, we found that the nodal straight lines can be classified into two types. One is the accidental degeneracy, nodal straight lines located in Brillouin zone (BZ), such as nodal straight lines in Rb 2 Sn 2 O 3 [37]. Its matrix elements of Hamiltonian vanish without physical reasons. The other is the symmetry-enforced degeneracy. Nodal straight lines locate on the high-symmetry lines of BZ, such as nodal straight lines in MgB 2 [36]. However, the ideal nodal straight lines have not been discovered in BZ of the realistic materials. The physical properties of nodal-straight-line semimetals also need to be further explored, which is important for future experimental probes and possible applications.
In this work, based on the first-principles calculations and symmetry analyses, we theoretically proposed that the quaternary-compounds CsNb 3 SBr 7 is a topological nodal-straight-line semimetal candidate. Due to the protection of crystal symmetries, bands are inevitably degenerate and form two parallel nodal straight lines located in the k z = 0 plane of BZ. The nodal straight lines bring nontrivial topological properties and exotic belt-like surface states on the (001) surface of CsNb 3 SBr 7 , which can be verified by the angle-resolved photoemission spectroscopy (ARPES) experiments. Furthermore, the Hofstadter's butterfly and optical conductivity of CsNb 3 SBr 7 are investigated using a slab sample, which is important for the further condensed matter physics experiments.

Crystal structure and computational methods
Quaternary-compounds CsNb 3 SBr 7 crystallizes in monoclinic crystal structure with the nonsymmorphic space-group P2 1 /c (No. 14). Its lattice constants are a = 10.38Å, b = 7.16Å and c = 19.57Å. The angles between crystal axes are α = 90 • , β = 90 • , and γ = 82.31 • , respectively [39]. The primitive cell of CsNb 3 SBr 7 including 48 atoms is shown in figure 1(a). Clusters of Nb atoms are linked by Nb-S bonds, forming an infinite chains. The adjacent chains are bridged by Cs and Br atoms. The first BZ of CsNb 3 SBr 7 crystal is illustrated in figure 1(b), which is used in following discussions.
The numeral calculations are performed by Vienna ab initio simulation package (VASP) [40,41] with the generalized gradient approximation of Perdew-Burke-Ernzerhof type exchange-correlation potential [42,43]. The cut-off energy is set to 500 eV, and the maximum force is set to 0.01 eV. A 9 × 13 × 7 Monkhorst-Pack k-point grid [44] is used for the self-consistent calculations. The nonlocal Heyd-Scuseria-Ernzerhof (HSE) hybrid functional calculations [45] are also used to check the band structure. The irreducible representations of bands are calculated using the IRVSP program in conjunction with VASP [46]. The tight-binding model is obtained by constructing maximally localized Wannier functions (MLWF) [47,48]. Topological surface states are calculated by using iterative Green function method [49] as implemented in the WANNIERTOOLS package [50]. The Hofstadter's butterfly and optical conductivity of CsNb 3 SBr 7 were calculated using the tight-binding propagation method (TBPM) [51], which is based on the numerical solution of the time-dependent Schrödinger equation without the diagonalization of the Hamiltonian matrix.

Electronic structure and topological properties
The band structure of CsNb 3 SBr 7 without SOC is presented in figure 2(a). One observes that conduction and valence bands touch in Γ-A and Γ-B high-symmetry lines, indicating CsNb 3 SBr 7 is a semimetal. Figure 2(b) is the enlarged view of band structure along the Γ-A direction. Through calculating the irreducible representations of bands, we found that two crossing bands belong to the one-dimensional representation Γ 1 and Γ 2 of C s point group [52], respectively. According to the Schur's lemma, the two bases belonging to two different irreducible representations are orthogonal to each other. Hence, the two bands cross without opening a gap, resulting in a crossing point band degeneracy. Figure 2(c) is the band structure of CsNb 3 SBr 7 with SOC. Comparing to the result in figure 2(b), we found the crossing bands open an energy gap of 5 meV under the effect of SOC. Although SOC can induce the crossing bands to open energy gaps, the gap is too small to be measured by the ARPES experiment. Therefore, the effect of SOC can be ignored, and CsNb 3 SBr 7 is viewed as a nodal-straight-line semimetal. Similar situations also arose in nodal line semimetal ZrSiS that had been experimentally confirmed [53].
Next, we calculate the three-dimensional band dispersion of CsNb 3 SBr 7 in the k z = 0 plane of BZ. The result is shown in figure 3(a). We find that a band crossing appears in the momentum space, which implies that gapless points form the nodal straight lines in CsNb 3 SBr 7 . To clearly illustrate that the nodal straight lines are straight, we plotted the distribution of energy gap between two crossing bands in k z = 0 plane of BZ, as shown in figure 3(b). The red straight lines indicate the location of gapless points. Different from the nodal straight lines in Rb 2 Sn 2 O 3 [37], the gapless points accidentally form the very straight nodal lines in CsNb 3 SBr 7 . In order to clarify the topological property of nodal straight lines, we calculated the Berry phase where A n (k) = i ψ n (k)|∇ k |ψ n (k) is the Berry connection. ψ n (k) is Bloch wave function of the occupied states. l is a closed path along the k z direction due to the periodicity of BZ. In the one-dimensional parameterized systems, the Berry phase equals π for k inside the nodal straight lines, while it equals zero for k outside the nodal straight lines, as shown in figure 3(c). It clearly shows that the Berry phase gets a jump of π along the -B-Γ-B path. Furthermore, we have calculated the Berry curvature distributions of nodal straight lines in CsNb 3 SBr 7 . Berry curvature is calculated by the following equation [54]:

Symmetry analyses of the degenerated bands
The existence of topological nodal straight lines is intimately connected with crystal symmetries, raising the following question: what is the symmetry required for the nodal straight lines appear in BZ of CsNb 3 SBr 7 ?
Beside identity E, CsNb 3 SBr 7 only contains four symmetry operations: inversion P, time-reversal T and two nonsymmorphic symmetries: a glide mirror symmetry about the xy plane, i.e., M z = M z |0, 1 2 , 1 2 , and a twofold rotation symmetry along z axis, i.e., C 2z = 2 001 |0, 1 2 , 1 2 which are the products of symmetry operators and fractional translational vectors in the unit of the lattice constant. We first consider the M z symmetry, which has the invariant subspace in the k z = 0 plane. After applying M z twice, we note that the x and z coordinates keep invariant, but y coordinate will be shifted by one lattice spacing. As a result, M z 2 |ψ k = e −ik y ·b |ψ k for each Bloch state in the k z = 0 plane. When k y = ±π/b, the M z 2 = −1, which generates the Kramers-like degeneracy. In other words, for any energy eigenstate |ψ with M z eigenvalue g x , it must have a degenerate partner M z |ψ with M z eigenvalue −g x . Next, we turn to investigate the band-crossing points that are not isolated and form the nodal straight lines in BZ, as illustrated in figure 3(b). Because each state is double degeneracy at k y = ±π/b, and the degenerate partners have opposite M z eigenvalues ±g x , which can be labeled as (+, −). On the other hand, the corresponding four states are not required to be degenerate at Γ, where the M z eigenvalues are (−, −, +, +) for the states in ascending order. Focusing on the two middle bands, they have opposite M z eigenvalues, and their ordering is inverted. As a result, they must cross, forming the nodal straight lines in BZ. Besides, the combined operation C 2z T bring in an additional band degeneracy in the k z = ±π/c plane. Due to the periodic boundary condition, the momentum in k z = ±π/c plane is invariant under C 2z T operation. After applying C 2z T twice, the x and y coordinates keep invariant and z coordinate will be shifted by one lattice spacing. As a result, ( C 2z T) 2 |ψ k = e −ik z ·c |ψ k for each Bloch state in k z = ±π/c plane. Consequently, ( C 2z T) 2 = −1 for any k point in the k z = ±π/c plane, indicating the Kramers-like double degeneracy. These symmetry analyses agree well with the results originated first-principles calculations.

Topological surface states and Fermi arcs
The nontrivial topological properties of nodal straight lines are always accompanied by the exotic surface states and Fermi arcs. The momentum-resolved surface density of states (DOS) along the high-symmetry lines of (001) surface BZ is shown in figure 4(a). It is clearly visible that the bulk conductivity state and valence state touch in the high-symmetry line Γ-B, and a surface state connects the projections of two nodal points. Different from conventional drum-like surface state, the nontrivial nodal straight lines own belt-like surface state [36]. To understand the belt-like surface state, we plot the isoenergy plane of surface state spectrum at the Fermi energy, as shown in figure 4(b). It can be viewed as countless broken Fermi arcs nest between the nodal straight lines with different chirality. In addition, the corresponding surface state and Fermi arcs on the (100) surface of CsNb 3 SBr 7 have been shown in figures 4(c) and (d). These results can be verified by ARPES experiments.

Electronic and optical properties
In order to investigate the electronic and optical properties of CsNb 3 SBr 7 , we constructed a 20-unit-cells-thick slab sample with open boundary conditions in the z direction, containing 500 ×500× 20 cells. The DOS is calculated by Fourier transform of the time-dependent correlation functions in which, the Hamiltonian is originated from the maximally localized WFs, the time evolution operator e −iHt is obtained by standard Chebyshev polynomial representation, and the |ψ is a random superposition of all the basis states in real space, i.e., where a i are random complex numbers normalized as i |a i | 2 = 1. The bulk DOS is shown in figure 5(a). One observes that there are four Van Hove singularities at E = −0.33 eV, −0.03 eV, 0.032 eV and 0.54 eV, respectively. These singularities correspond to the flat band structure in figure 2(a). The surface DOS of slab CsNb 3 SBr 7 is shown in figure 5(b). Comparing the bulk and surface densities of states, the significant difference is the emergence of Van Hove singularities in the vicinity of Fermi level, which is attributed to the nodal-straight-line surface state. These results can also be reproduced using the MLWF method. Then, we extend the calculation of surface DOS under an external magnetic field which is perpendicular to the slab plane. The hopping terms t ij of Hamiltonian are replaced by the Peierls substitution where Φ 0 = hc/e is the flux quantum and A = (−B y , 0, 0) is the vector potential in the Landau gauge. The Hofstadter's butterfly under the magnetic field less than 50 tesla (T) is shown in figure 5(c). An obvious zero-mode landau level appears at the Fermi energy, which is ascribed to the topological surface state. Meanwhile, the zero-mode landau level shows a higher DOS as the strength of magnetic field increases. In the vicinity of low energies, the energy states show quantized behavior, while the quantized phenomenon becomes unconspicuous at higher energies. This is because the highly localized DOS appears around the energy E = ±0.03 eV, which efficiently affects the properties of landau level under magnetic field. We expect that the CsNb 3 SBr 7 exhibits interesting magnetoresponses which is distinct from the usual semimetal materials [55]. Then, we investigated the optical properties of CsNb 3 SBr 7 . The Drude contribution is ignored at ε = 0 and the energy-dependent optical conductivity at 300 K is calculated by using the Kubo formula where β = 1/k B T is the inverse temperature, Ω is the sample area, f(H) = (e β(H−μ) + 1) −1 is the Fermi-Dirac distribution operator, J α (t) = e iHt J α e −iHt is the time-dependent current operator in the α direction. The time evolution operator e −iHt and the Fermi-Dirac distribution operator f (H) are obtained by the standard Chebyshev polynomial representation. The optical conductivity of CsNb 3 SBr 7 is shown in figure 5(d). It is well known that the optical conductivity is closely related to the DOS of occupied and unoccupied states. Through comparing the numerical results of optical conductivity and DOS of CsNb 3 SBr 7 , we identified the one-to-one correspondence of peaks in optical spectrum and DOS, which involved the particle-hole excitation between valence and conduction bands. In the high-energy part of optical spectrum, a peak is noticeable at energy E = 0.87 eV, which is associated with optical transitions between the Van Hove singularities of DOS at energies E = −0.33 eV and E = 0.54 eV. Besides, an additional peak appears in the low-energy area, which is associated with optical transitions between the Van Hove singularities of DOS at energies E = −0.03 eV and E = 0.032 eV. These characteristics are similar to the graphene in low energy [56], which can be measured in the future optical experiments.

Summary
In summary, we theoretically propose that the quaternary-compounds CsNb 3 SBr 7 is a nodal-straight-line semimetal candidate through the first-principles calculations and symmetry analyses. Due to the protection of glide symmetry M z , two parallel nodal straight lines locate in the k z = 0 plane of BZ. The topological properties of nodal straight lines are characterized by the nontrivial Berry phase and Berry curvature. On the (001) surface of CsNb 3 SBr 7 , Fermi arcs form the belt-like surface states, connecting the nodal straight lines with opposite chirality. Furthermore, the Hofstadter's butterfly and optical conductivity of CsNb 3 SBr 7 are investigated by using a slab sample. We find that nodal-straight-line state brings a highly localized surface DOS at Fermi level. Under the effect of perpendicular magnetic fields, the energy states show quantized behavior at low energies. The sharp peak of optical conductivity mainly originated from the optical transitions between the Van Hove singularities of bulk DOS in CsNb 3 SBr 7 . These results are important for future theoretical and experimental studies on nodal-straight-line semimetal materials.